TITLE: Phase Retrieval: Uniqueness Guarantees, Tensorial Liftings, and Applications

ABSTRACT: Recovering an unknown signal from phaseless data is the essential task behind every phase retrieval problem. Nowadays phase retrieval is a general concept with a wide range of possible applications, a well-developed theory, and numerous numerical solvers.  From a mathematical perspective, the crucial task is to recover an unknown function from phaseless intensity measurements of its Fourier transform. Although the Fourier transform is a well-understood operator, the loss of the phase turns the reconstruction task into a severely ill-posed inverse problem, where the strong ambiguousness ranks first of the analytic and algorithmic challenges.  The aim of the talk is to give an overview over the phase retrieval problem, introduce some modern applications like phase retrieval in optical diffraction tomography and dynamical sampling, and to discuss strategies to overcome the ambiguousness especially in super-resolution phase retrieval.  Many phase retrieval problems may be interpreted as bilinear or quadratic inverse problems opening the way to use tensorial liftings to develop an adapted regularization theory.  These liftings may also be used to derive numerical solvers based on convex optimization.  Moreover, the resulting algorithms can be implemented in a tensor-free manner.